= Supremum
{title2=$\sup$}
= Suprema
{synonym}
For a nonempty <set> $A\subseteq\mathbb R$, a real number $s$ is its <supremum> precisely when it is an <upper bound> and no smaller real number is an <upper bound>. It is unique: two least <upper bounds> must each be no greater than the other.
For $A=(0,1)$, every number at least one is an <upper bound>, and none smaller than one bounds the whole <set>. Thus $\sup A=1$, although $1\notin A$. For $A=[0,1]$, that same <supremum> is also the <maximum>. The <approximation property of a supremum> supplies witnesses arbitrarily close to an unattained upper edge.
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